Re: Cantor and the binary tree



Martin Shobe wrote:
> On 16 Jul 2005 08:28:50 -0700, mueckenh@xxxxxxxxxxxxxxxxx wrote:
>
> >
> >
> >Martin Shobe wrote:
> >
> >> Nope. Here the sequence converges (you've admitted that), and the
> >> value it converges to is clearly not on the list.
> >
> >0.000...
> >0.1000...
> >0.11000...
> >0.111000...
> >...
> >
> >If in this list the exchange prescription is 0 --> 1, then the
> >antidiagonal is Sum{n = 1 to oo} 10^-n = 1/9. This number is not a
> >member of the list. But there can be set up a bijection between the
> >digits of the antidiagonal and the digits of the lines. This shows that
> >1/9 is not on the diagonal unless it is in a line.
>
> No it doesn't. This is the same fallicious argument you make about N.
> Just becuase something is true for every prefix if 1/9 (initial
> segment of N) it does not follow that it is true for 1/9 (N). 1/9 (N)
> simply does not meet the conditions in your proof.

Can you state a difference larger than zero between 1/9 and a line
number for ALL line numbers? (Not for the n-th, but for ALL.)
>
> > Only intermingling
> >limits has lead to that false conclusion which is considered foundation
> >of mathematics for more than 100 years now.
>
> This is laughable, since that is exactly what you have been doing, and
> the mathematicians have been avoiding.

Try to answer the question above and you will see where mathematics
failed.
>
> >> Cantor's proof goes
> >> through. You have yet to prove whether or not your transpositions
> >> converge (you have yet to fix a topology to make the question of
> >> convergance even meaningful), and once you do that, you still need to
> >> prove that the order it converged to is both well-orderd and respects
> >> the standard order of the naturals.
> >
> >According to Cantor, the well-order remains a well-order after
> >countably many transpositions. I just believe him in this respect.
>
> I don't know what he wrote here, so I can only say the following.
>
> Either Cantor was wrong about this, or (more likely) WM has
> misunderstood him.

Here are two German texts by Cantor. One saying that number of elements
(Anzahl) depends on the well-order. The second saying that Anzahl is
not changed by transpositions.

Unter Anzahl einer wohlgeordneten Menge M verstehe ich den
Algemeinbegriff (Gattungsbegriff), welchen man erhält, indem man bei
der wohlgeordn. Menge M von der Beschaffenheit und Bezeichnung ihrer
Elemente abstrahiert und nur auf die Rangordnung reflectirt, durch
welche die Elemente miteinander verbunden sind; die Anzahl einer Menge
ist also allen Mengen desselben Typus gemeinsam, gewissermaaßen
dasjenige, was ihnen allen immanent ist. Diesen nämlichen
Allgemeinbegriff können wir auch Zahl nennen und zwar diejenige Zahl,
unter welcher die gegebene wohlgeordnete Menge steht oder die zu der
die wohlgeordnete Menge M gehört.

Die Frage, durch welche Umformungen einer wohlgeordneten Menge ihre
Anzahl geändert wird, durch welche nicht, läßt sich einfach so
beantworten, daß diejenigen und nur diejenigen Umformungen die Anzahl
ungeändert lassen, welche sich zurückführen lassen auf eine endliche
oder unendliche Menge von Transpositionen, d. h. von Vertauschungen je
zweier Elemente.

Regards, WM

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